Rota-Baxter operators on dihedral and alternating groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916217379356672 |
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| author | Galt, Alexey Gubarev, Vsevolod |
| author_facet | Galt, Alexey Gubarev, Vsevolod |
| contents | Rota-Baxter operators on algebras, which appeared in 1960, have connections with different versions of the Yang-Baxter equation, pre- and postalgebras, double Poisson algebras, etc. In 2020, the notion of Rota-Baxter operator on a group was defined by L. Guo, H. Lang, Yu. Sheng.
In 2023, V. Bardakov and the second author showed that all Rota-Baxter operators on simple sporadic groups are splitting, i. e. they are defined via exact factorizations. In the current work, we clarify for which $n$, there exist non-splitting Rota-Baxter operators on the alternating group $\mathrm{A}_n$. For the corresponding $n$, we describe all non-splitting Rota-Baxter operators on $\mathrm{A}_n$. Moreover, we describe Rota-Baxter operators on dihedral groups $D_{2n}$ providing the general construction which lies behind all non-splitting Rota-Baxter operators on $\mathrm{A}_n$ and $D_{2n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14078 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rota-Baxter operators on dihedral and alternating groups Galt, Alexey Gubarev, Vsevolod Group Theory Rota-Baxter operators on algebras, which appeared in 1960, have connections with different versions of the Yang-Baxter equation, pre- and postalgebras, double Poisson algebras, etc. In 2020, the notion of Rota-Baxter operator on a group was defined by L. Guo, H. Lang, Yu. Sheng. In 2023, V. Bardakov and the second author showed that all Rota-Baxter operators on simple sporadic groups are splitting, i. e. they are defined via exact factorizations. In the current work, we clarify for which $n$, there exist non-splitting Rota-Baxter operators on the alternating group $\mathrm{A}_n$. For the corresponding $n$, we describe all non-splitting Rota-Baxter operators on $\mathrm{A}_n$. Moreover, we describe Rota-Baxter operators on dihedral groups $D_{2n}$ providing the general construction which lies behind all non-splitting Rota-Baxter operators on $\mathrm{A}_n$ and $D_{2n}$. |
| title | Rota-Baxter operators on dihedral and alternating groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2404.14078 |